A Fast Algorithm for Sparse Reconstruction Based on Shrinkage, Subspace Optimization, and Continuation

We propose a fast algorithm for solving the ℓ1-regularized minimization problem minx∈Rn µ x 1 + Ax − b 2 2 for recovering sparse solutions to an undetermined system of linear equations Ax = b. The algorithm is divided into two stages that are performed repeatedly. In the first stage a first-order iterative method called “shrinkage” yields an estimate of the subset of components of x likely to be nonzero in an optimal solution. Restricting the decision variables x to this subset and fixing their signs at their current values reduces the ℓ1-norm x 1 to a linear function of x. The resulting subspace problem, which involves the minimization of a smaller and smooth quadratic function, is solved in the second phase. Our code FPC AS embeds this basic two-stage algorithm in a continuation (homotopy) approach by assigning a decreasing sequence of values to µ. This code exhibits state-of-the-art performance both in terms of its speed and its ability to recover sparse signals. It...

• Type: Journal
• Year: 2010
• Where: SIAMSC
• Authors: Zaiwen Wen, Wotao Yin, Donald Goldfarb, Yin Zhang
• Keywords: Linear Equations Ax, SIAMSC 2010, Sparse, ℓ1-regularized Minimization Problem